Estimates on the difference between succeeding eigenvalues and Lifshitz tails for random Schrödinger operators

1987 ◽  
pp. 138-151 ◽  
Author(s):  
Werner Kirsch
Keyword(s):  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
Lifshitz Tails ◽  
The Difference

scholarly journals THE DENSITY OF STATES AND THE SPECTRAL SHIFT DENSITY OF RANDOM SCHRÖDINGER OPERATORS

2000 ◽  
Vol 12 (06) ◽  
pp. 807-847 ◽  
Author(s):  
VADIM KOSTRYKIN ◽  
ROBERT SCHRADER
Keyword(s):  
Density Of States ◽  
Arbitrary Dimension ◽  
Schrödinger Operators ◽  
Spectral Shift ◽  
Random Schrödinger Operators ◽  
Shift Function ◽  
Schrodinger Operators ◽  
Spectral Shift Function ◽  
Arbitrary Dimensions ◽  
The Difference

In this article we continue our analysis of Schrödinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in arbitrary dimensions. For arbitrary dimension we show existence of the spectral shift density, which is defined as the bulk limit of the spectral shift function per unit interaction volume. This density equals the difference of the density of states for the free and the interaction theory. This extends the results previously obtained by the authors in one dimension. Also we consider the case where the interaction is concentrated near a hyperplane.

scholarly journals Internal Lifshitz tails for discrete Schrödinger operators

2006 ◽  
Vol 2006 ◽  
pp. 1-8
Author(s):  
Hatem Najar
Keyword(s):  
Density Of States ◽  
Present Model ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Integrated Density Of States ◽  
Periodic Operator ◽  
Schrodinger Operators ◽  
Spectral Gaps ◽  
Lifshitz Tails ◽  
Discrete Schrödinger Operators

We consider random Schrödinger operatorsHωacting onl2(ℤd). We adapt the technique of the periodic approximations used in (2003) for the present model to prove that the integrated density of states ofHωhas a Lifshitz behavior at the edges of internal spectral gaps if and only if the integrated density of states of a well-chosen periodic operator is nondegenerate at the same edges. A possible application of the result to get Anderson localization is given.

scholarly journals Supersymmetric Cluster Expansions and Applications to Random Schrödinger Operators

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Luca Fresta
Keyword(s):  
Density Of States ◽  
Local Density ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
Local Density Of States ◽  
Weak Disorder ◽  
Cluster Expansions ◽  
Strong Disorder ◽  
Lifshitz Tail

AbstractWe study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy. As an application, we establish Lifshitz-tail-type estimates for the local density of states and thus localization at weak disorder.

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